In Physics, hot is an adjective meaning at high temperature. Heat is a different concept. In the case of ordinary matter, the temperature can be associated with atomic speeds. However, it is possible to generalize the concept of temperature to deal with systems different from moving particles.
The most general thermodynamical definition of temperature is based on the rate of change with the energy of the number of the microscopic states of a system. Quite an abstract definition but sufficiently general to allow using the word temperature in cases where the classical concept of atomic movement cannot be used (quantum systems, or electromagnetic fields, to cite a couple of examples).
In the context of the Big Bang, the concept of density of the universe also needs a generalization of the usual idea of the mass per unit volume. The proper context for Big Bang theory is General Relativity (GR). Relativity tells us that mass and energy are not separate concepts: an increase in energy implies an increase in mass. Therefore, extremely dense is equivalent to saying that a huge amount of energy was confined in a very small volume.
Unfortunately, in many cases, the popularization of Science uses common language words without explaining that they may have a different meaning in Physics.